SPPU Engineering Maths 3 - May 2017 Exam Question Paper

SPPU Electronics and Telecom Engineering (Semester 4)
Engineering Maths 3
May 2017

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

Solve any two question Q.1(a)(i, ii, iii) Solve any one question from Q.1(a,b) & Q.2(a,b,c)

1(a)(i)

(D^{2} - 7 D + 6) y = e^{2 x}

$\left ( D^{2}-7D+6 \right )y=e^{2x}$

4 M

1(a)(ii)

(D^{2} + 4) y = x sin x

$\left ( D^{2} +4\right )y = x\sin x$

4 M

1(a)(iii)

{(x + 2)}^{2} \frac{d^{2} y}{d x^{2}} + 3 (x + 2) \frac{d y}{d x} + y = 4 sin log (x + 2)

$\left ( x+2 \right )^{2}\frac{d^{2}y}{dx^{2}}+3\left ( x+2 \right )\frac{dy}{dx}+y=4\sin \log \left ( x+2 \right )$

4 M

1(b) Find the Fourier sine transform of the function:

f (x) = e^{- x}, x > 0.

$f(x)=e^{-x},x> 0.$

4 M

2(a) A circuit consists of an inductance L and condenser of capacity C in series. An alternating e.m.f. E sin n t is applied to it at time t = 0, the initial current and charge on the condenser being zero and

w^{2} = \frac{1}{L C},

$w^{2}=\frac{1}{LC},$ / find the current flowing in the circuit at any time for w≠n.

4 M

Solve any two question Q.2(b)(i, ii)

2(b)(i) Find inverse z-transform

F (z) = \frac{z}{(z - 1) (z - 3)}, | z | > 3

$F(z)=\frac{z}{\left ( z-1 \right )\left ( z-3 \right )},|z|> 3$

4 M

2(b)(ii)

F (z) = \frac{z}{(z - \frac{1}{4}) (z - \frac{1}{5})} .

$F(z)=\frac{z}{\left ( z-\frac{1}{4} \right )\left ( z-\frac{1}{5}\right )}.$ / (by inversion integral method).

4 M

2(c) Solev the following difference equation:

$\begin{align*}f\left ( k+1 \right )+\frac{1}{4}f(k)=\left ( \frac{1}{4} \right )^{k};\\ f(0)=0, k\geq 0 \end{align*}$ /

4 M

Solve any one question from Q.3(a,b,c) &Q.4(a,b,c)

3(a) Using fourth order Runge-Kutta method, solve the differential equation:

$\frac{dy}{dx}=x+y+xy$ / with y(0) = 1 to get y(0, 1) taking h = 0.1

4 M

3(b) Find Lagrange's interpolating polynomial passing through the set of points:

x	y
0	3
1	4
3	12

4 M

3(c) Find the directional derivative of:

$\phi =x^{2}-y^{2}+2z^{2}$ / at the point (1, 2, 3) in the direction of

$4\bar{i}-2\bar{j}+\bar{k}.$

4 M

Solve any two question Q.4(a)(i, ii)

4(a)(i) Show that:

$\nabla\left ( \frac{\bar{a}.\bar{r}}{r^{2}} \right )=\frac{\bar{a}}{r^{2}}-\frac{2\left ( \bar{a}.\bar{r} \right )\bar{r}}{r^{4}}$

4 M

4(a)(ii)

$\nabla\left ( \bar{a} .\nabla\frac{1}{r}\right )=\frac{3\left ( \bar{a}.\bar{r} \right )\bar{r}}{r^{5}}-\frac{\bar{a}}{r^{3}}$

4 M

4(b) Show that :

$\bar{F}=\left ( 6xy+z^{3} \right )\bar{i}+\left ( 3x^{2}-z \right )\bar{j}+\left ( 3xz^{2} -y\right )\bar{k}$ / is irrotational. Find the scalar pothential ϕ such that

$\bar{F}=\nabla\phi .$

4 M

4(c) By using Trapezoidal rule, evaluate:

$\int_{0}^{2}\frac{1}{1+x^{4}}\ dx$ / taking

$h=\frac{1}{2},$ / correct upto 3-decimal places.

4 M

Solve any two question from Q.5(a,b,c) & Solve any one question from Q.5(a, b,c) &Q.6(a, b, c)

5(a) Evaluate

$\int \bar{F}.d\bar{r}\ \text{bar}\\ \ \bar{F}=3x^{2}\ \bar{i}+\left ( 2xy-y \right )\bar{j}+z\bar{k}$ / also straight line joining (0, 0, 0) and (2, 1, 3).

4 M

5(b) Evaluate :

$\iint _{S}\left [ \left ( 4x+3yz^{2} \right ) \bar{i}-\left ( x^{2}z^{2} +y\right )\bar{j}+\left ( y^{2} +2z\right )\bar{k}\right ].d\bar{S}$ / where S is the surface of the sphere

$x^{2}+y^{2}+z^{2}=9.$

4 M

5(c) Apply Stokes's theorem to evaluate

$\oint _{C}\bar{F}.d\bar{r}$ / where:

$\bar{F}=xy^{2}\bar{i}+y\bar{j}+xz^{2}\bar{k}$ / and C is the boundary of reactangle:
x=0, y=0, x=1, y=2 in z=0 plane.

5 M

6(a) Using Green's lemma evaluate:

$\int _{C}\left ( xy-x^{2} \right )dx+x^{2}y \ \ dy$ / along the curve C : x=1, y=x, y=0.

4 M

6(b) Evaluate:

$\iint \left ( \nabla\times \bar{F} \right ).\hat{n} \ \ ds,$ / where

$\bar{F}=\left ( x-y \right )\bar{i}-\left ( x^{2}+yz \right )\bar{j}-3xy^{^{2}}\bar{k}$ / and S is the surface of the cone

$z=4-\sqrt{x^{2}+y^{2}},$ / above the xoy plane.

5 M

6(c) Prove that:

$\iint _{s}\left ( \phi \nabla\Psi -\Psi \nabla\phi \right ).d\bar{S}=\iint _{v}\int \left ( \phi \nabla^{2} \Psi -\Psi \nabla^{2}\phi \right )dV.$

4 M

Solve any one question from Q.7(a,b,c) & Q.8(a,b,c)

7(a) Show that the function:

$f(z)= u+iv$ / with constant modulus and constant amplitude is constant in each case.

4 M

7(b) Evaluate:

$\oint _{C}\frac{4z^2+z}{z^{2}-1}\ \ dz$ / where C is the circle |z-1| =

$\frac{1}{2}$ .

4 M

7(c) Find the bilinear tranformation which maps the points: z=1, i, 2i onto the points w = -2i, 0, 1 respecitvely.

5 M

8(a) If f(z) = u+iv is analytic, find f(z), if

$u-v= x^{2}-y^{2}-2xy.$ /

5 M

8(b) Evaluate:

$\oint _C\frac{\left ( z^{2} +cos^{2}z\right )}{\left ( z-\frac{\pi }{4} \right )}^{3} dz,$ / where C is circle |z| =1

4 M

8(c) Find the map of the straight line y=x under the transformation

$w=\frac{z-1}{z+1}.$

4 M

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